A) \[f\left( f\left( x \right) \right)=\text{ }x\] \[\operatorname{for}\,some\]\[x\in R\]
B) \[f\left( f\left( x \right) \right)>x\,\,\forall \,\,x\in R\]
C) \[f\left( f\left( x \right) \right)<x\,\,\forall \,\,x\in R\]
D) None of these
Correct Answer: B
Solution :
| Let \[g\left( x \right)=f\left( x \right)-x,\] so \[g\left( x \right)\] is continuous and \[g\left( 0 \right)=1.\] |
| Now it is given that \[g\left( x \right)\ne 0\]for any\[x\in R\] |
| So, \[g\left( x \right)>0\,\forall \,x\in R\,\operatorname{i}.e.f\left( x \right)>x\,\forall \,x\in R\]s \[\Rightarrow \]\[f\left( f\left( x \right) \right)>f\left( x \right)>x\forall x\in R.\] |
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